Acceleration in One, Two, and Three Dimensions in Launched Roller Coasters
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S PECIAL F EATURE: E XTREME P HYSICS www.iop.org/journals/physed Acceleration in one, two, and three dimensions in launched roller coasters Ann-Marie Pendrill Department of Physics, G¨oteborg University, SE 412 96 G¨oteborg, Sweden E-mail: [email protected] Abstract During a roller coaster ride, the body experiences acceleration in three dimensions. An accelerometer can measure and provide a graph of the forces on the body during different parts of a ride. To couple the experience of the body to pictures of the ride and an analysis of data can contribute to a deeper understanding of Newton’s laws. This article considers the physics of launched roller coasters. Measurements were performed with a three-dimensional co-moving accelerometer. An analysis is presented of the forces in the different ride elements of the Kanonen in G¨oteborg and the Speed Monster in Oslo, which both include loops and offer rich examples of force and acceleration in all dimensions. Introduction The Stealth is the highest of the European 3, 2, 1 ... launch! The traditional lift hill, which launch coasters. After the launch, the train passes gives the initial potential energy for the ride, is the ‘top hat’ (figure 3) and then returns over a absent in some newly built roller coasters. Instead, camel back into a hairpin turn back into the station. the initial energy is provided in the form of a The Kanonen and Speed Monster roller horizontal launch, giving sufficient kinetic energy coasters both feature a loop and a screw during the to bring the train to the top of the first hill. ride. The accelerometer data and elevation profile From then on, the ride is characterized by the from these rides are shown in figures 4 and 5,and interchange between potential and kinetic energy, discussed in more detail below. in the same way as in traditional roller coasters. The first Intamin hydraulic launch coaster in One-dimensional horizontal motion Europe was Rita the Ride at Alton Towers, which In schools, the study of motion traditionally starts opened in April 2005, followed two weeks later by with non-motion, continuing with motion in one Kanonen at Liseberg in G¨oteborg (figure 1). The dimension. The traditional lift hill is an example Speed Monster at Tusenfryd in Oslo (figure 2)and of uniform rectilinear motion, where Newton’s the Stealth at Thorpe Park (figure 3) both opened first law applies. The launch is an example of in 2006. In 2007, similar launch coasters were accelerated motion in one dimension—as is the added to Heide-Park in Germany and PortAventura final brake. These situations can be useful as in Spain [1–3]. (See also the Roller Coaster Data illustrations to textbook presentations. In one Base at www.rcdb.com.) dimension, the measurement of the acceleration 0031-9120/08/050483+09$30.00 © 2008 IOP Publishing Ltd P HYSICS E DUCATION 43 (5) 483 A-M Pendrill Figure 1. The Kanonen roller coaster viewed from the side, showing the launch from the left into the ‘top hat’ on the right, as well as the shape of the clothoid loop. Figure 2. Panorama of the Speed Monster. The launch is from the right into the Norwegian loop, which encircles the entrance escalator. (Photo: Jochen Peschel [1].) in the direction of motion gives full information nitrogen is compressed to a pressure of around about the motions if the initial speed is known. 300 bar. During launch, the gas is allowed to The variation of speed and distance with time is expand rapidly, sending the hydraulic oil through obtained by integration, which can be performed the motors, and energy is transferred to the numerically or analytically, after approximation of accelerating roller coaster. The technique is the acceleration time dependence. described in some detail by Peschel [3], who also presents an animation of the launch process. The pressure drops to about 250 bars, consistent The launch with the drop in horizontal acceleration during the Flags in the launch area enhance the sensation launch, seen from the graphs in figures 7 and 8. of motion during launch of the Speed Monster, Figures 7 and 8 shows the accelerometer as shown in figure 6. Horizontal launches of data for the Kanonen and Speed Monster rides. roller coasters have been used since the 1970s: The graphs also include speed and distance, for example, in the Revolution [2]whichisa obtained by numerical integration. From the Schwarzkopf ‘shuttle launch coaster’ [4], where graphs in figures 7 and 8, we can conclude that the energy is stored in a flywheel. Magnetic launch the force drops during the launch. This is natural techniques were introduced during the 1990s, with since the pressure of the nitrogen would drop as LIMs (linear induction motors) and LSMs (linear the gas expands, as discussed below. A fully synchronous motors). The compressed air launch loaded Kanonen train with four cars weighs about was introduced in 2002, followed by the hydraulic 8 tonnes. The Speed Monster train with three cars launch in 2002. The hydraulic launch was used is lighter, about 6 tonnes. These weights include to break a new altitude record in 2003 for the Top the mass of the sled used during acceleration. Thrill Dragster at Cedar Point, Ohio [2, 5]. Exercises for the reader. In the hydraulic launch, oil is pumped • What average power is needed to accelerate from a reservoir into storage cylinders filled the trains (in W and horsepower with nitrogen. The energy is built up as the (1 hp = 735 W))? 484 P HYSICS E DUCATION September 2008 Acceleration in one, two, and three dimensions in launched roller coasters 4 /g 2 vert a 0 0 5 10 15 20 25 30 35 40 45 t(s) 20 10 height (m) 0 0 5 10 15 20 25 30 35 t(s) Figure 5. Accelerometer and elevation data for the Speed Monster. Figure 3. The 62 m high ‘top hat’ of the Stealth roller coaster at Thorpe Park. 4 Figure 6. The launch of the Speed Monster, with a side /g tot view of the ‘Norwegian loop’. a 2 /g, vert 0 a –2 • How high above the starting point can the 0 5 10 15 20 25 30 35 t(s) Kanonen and Speed Monster trains go after launch? 20 10 Acceleration measurements in three height (m) 0 dimensions In roller coasters, as in everyday life, acceleration 0 5 10 15 20 25 30 35 t(s) is rarely restricted to one dimension. The forces Figure 4. Accelerometer and elevation data for required for the acceleration in a roller coaster are Kanonen. The green accelerometer curve shows evident throughout the body. What the body can magnitude of the ‘g-force’, whereas the blue curve shows only the vertical component. experience can also be measured with a co-moving sensor. Since the body moves in the gravitational field, g, from the Earth, the additional force per • How does the power, P, vary during the mass unit required to obtain an acceleration, a,is launch? (Remember that power is force times (a − g). What is measured by an accelerometer is velocity, P = Fv.) thus in general not acceleration, but one or more September 2008 P HYSICS E DUCATION 485 A-M Pendrill 30 ) 2 30 20 (m/s x 20 a 10 10 (m/s), 0 x v 0 0 1 2 (m), x 2 3 4 5 Figure 7. Horizontal acceleration (m/s2) for the t(s) Kanonen launch, together with velocity (m/s) and Figure 8. Launch of the Speed Monster: acceleration distance (m) obtained through numerical integration. (m/s2), velocity (m/s) and distance (s). Which graph is which? The drop in acceleration in figures 7 and 8 corresponds to a drop in force and thus to the drop in pressure during launch, which can be used to estimate the fraction of the maximum elements in a roller coaster, the lateral components possible work exerted by the gas during the Kanonen vanish if the curves are perfectly banked. and Speed Monster launches. A problem in measuring acceleration in three dimensions is to keep the sensor axis aligned components of this vector. Since the gravitational with the body axis. When the sensor is kept acceleration is used as a reference, it is natural to safe in a vest on the body, the z-axis tends give results in terms of the ratio (a − g)/g.This to slope slightly backwards and sometimes also expression can be taken as a vector definition of sideways. A mathematically simple option is to |a − g| the ‘g-force’. use the magnitude of the vector , possibly incorporating the sign from the dominating The accelerometer data in this paper were vertical component to maintain ‘negative g’ obtained using a wireless dynamic sensor system readings. The Kanonen data in figure 4 show a from Vernier. This system also measures the comparison of the total g-force and the vertical air pressure and converts the barometer data to component. provide indications of altitude during the ride. When the other coordinates are also of Through Bernoulli’s principle, the altitude data interest, as for launch, break and roll, it is are influenced by speed, thus leading to an necessary to perform a coordinate transformation. overestimate of altitude for high speeds. (This can The data in this paper were transformed by rotating be seen, for example, around launch in the graphs the axes so that the data have only a vertical in figures 4 and 5.) component before the ride starts, and assuming that the sensor orientation relative to the track is Coordinate system for amusement ride fixed.